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If |x - 1|>=0 ---->then the modulus will be equal to (x-1) & roots of the resulting equation will be 2,1If |x - 1|<0 ---->then the modulus will be equal to (-x+1) & roots of the resulting equation will be 4,1

So the product of all the roots (2,1,4,1) is 8.
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I have a question, I do understand that why have you taken the value 1 but I don't understand why have you taken x>=1. Why not simply x>1

x could be 1, thus when you consider the ranges you should include this value in either of the range, so we could consider x<1 and x>=1 OR x<=1 and x>1 (you cam include = sign in either of the ranges).

When \(x>{-2}\), then \(|x+2|=(x-2)\). So, in this case we'll have \(x^2 + 4x + 7 =(x + 2) + 3\) --> \(x=-2\) or \(x=-1\). The first solution is not valid since it's out of the range we consider. The second one is OK.